If $\displaystyle \alpha, \beta$ are the zeroes of the quadratic polynomial $\displaystyle \mathrm{px}^{2}+\mathrm{qx}+\mathrm{r}$, then find the value of $\displaystyle \alpha^{3} \beta+\beta^{3} \alpha$.

\(\displaystyle \mathrm{px}^{2}+\mathrm{qx}+\mathrm{r}\) \(\displaystyle \alpha+\beta=\frac{-\mathrm{q}}{\mathrm{p}}, \alpha \beta=\frac{\mathrm{r}}{\mathrm{p}}\) \(\displaystyle \alpha^{3} \beta+\beta^{3} \alpha\) \(\displaystyle =\alpha \beta\left(\alpha^{2}+\beta^{2}\right)\) \(\displaystyle =\alpha \beta\left[(\alpha+\beta)^{2}-2 \alpha \beta\right]=\frac{\mathrm{r}}{\mathrm{p}}\left[\left(-\frac{\mathrm{q}}{\mathrm{p}}\right)^{2}-2\left(\frac{\mathrm{r}}{\mathrm{p}}\right)\right]\) \(\displaystyle =\frac{\mathrm{r}}{\mathrm{p}^{3}}\left(\mathrm{q}^{2}-2 \mathrm{pr}\right)\)